In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets?

Vitali's Lemma: Let E be a set of finite outer measure and G a collection of intervals that cover E in the sense of Vitali. Then given e> 0 there is a finite disjoint collection of intervals in G such that $m*(e - \bigcup_{n=1}^N I_n) < e$.

I'm trying to learn this theorem and I keep replacing "outer measure" with "measure" and I want a reason to stop doing that.

In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets?

Vitali's Lemma: Let $E$ be a set of finite outer measure and $G$ a collection of intervals that cover $E$ in the sense of Vitali. Then given $\varepsilon> 0$ there is a finite disjoint collection of intervals in $G$ such that $m^*(E - \bigcup_{n=1}^N I_n) < \varepsilon$.

I'm trying to learn this theorem and I keep replacing "outer measure" with "measure" and I want a reason to stop doing that.